Problem 60 - Chemistry: The Central Science in SI Units, Global Edition

Question

Determine the \(\mathrm{pH}\) of each of the following solutions \(\left(K_{a}\right.\) and \(K_{b}\) values are given in Appendix D): (a) \(0.095 \mathrm{M}\) hypochlorous acid, \((\mathbf{b}) 0.0085 \mathrm{M}\) hydrazine, (c) \(0.165 \mathrm{M}\) hydroxylamine.

Step-by-Step Solution

Answer

Using the given initial concentrations and equilibrium constant expressions, the pH of each solution can be determined as follows: (a) Hypochlorous acid (acidic): Calculate [H+] using \(K_a = \frac{x^2}{0.095-x}\), then find pH: \(pH = -\log([H^+])\) (b) Hydrazine (basic): Calculate [OH-] using \(K_b = \frac{x^2}{0.0085-x}\), find pOH: \(pOH = -\log([OH^-])\), and convert to pH: \(pH = 14 - pOH\) (c) Hydroxylamine (basic): Calculate [OH-] using \(K_b = \frac{x^2}{0.165-x}\), find pOH: \(pOH = -\log([OH^-])\), and convert to pH: \(pH = 14 - pOH\)

Step 1: Identify whether the solution is acidic or basic

In this exercise, three solutions are given: (a) Hypochlorous acid: As the name suggests, it is an acid. (b) Hydrazine: This is a base. (c) Hydroxylamine: This is a base.

Step 2: Write down the ionization/dissociation reactions for each solution

For each of the given solutions, we need to write down their ionization or dissociation reactions, as follows: (a) Hypochlorous acid: \(HOCl \rightleftharpoons H^+ + OCl^-\) (b) Hydrazine: \(N_2H_4 + H_2O \rightleftharpoons N_2H_5^+ + OH^-\) (c) Hydroxylamine: \(NH_2OH + H_2O \rightleftharpoons NH_3OH^+ + OH^-\)

Step 3: Set up an equilibrium table and expressions for each solution

Now, we will set up an equilibrium table for each of these solutions, as well as the equilibrium constant expressions: (a) Hypochlorous acid: \(K_a = \frac{[H^+][OCl^-]}{[HOCl]}\) Initial concentration: [HOCl] = 0.095 M, [H+] = 0, [OCl-] = 0 (b) Hydrazine: \(K_b = \frac{[N_2H_5^+][OH^-]}{[N_2H_4]}\) Initial concentration: [N_2H_4] = 0.0085 M, [N_2H_5+] = 0, [OH-] = 0 (c) Hydroxylamine: \(K_b = \frac{[NH_3OH^+][OH^-]}{[NH_2OH]}\) Initial concentration: [NH_2OH] = 0.165 M, [NH_3OH+] = 0, [OH-] = 0

Step 4: Calculate [H+] or [OH-] using equilibrium constants

For each solution, refer to Appendix D and use the given equilibrium constant to find the concentration of [H+] or [OH-]. Solve the equations assuming x as the change in concentration: (a) Hypochlorous acid: As there is no initial [H+] or [OCl-], let x be the change. At equilibrium, [H+] = x, [OCl-] = x, and [HOCl] = 0.095 - x: \(K_a = \frac{x^2}{0.095-x}\) (b) Hydrazine: At equilibrium, [N_2H_5+] = x, [OH-] = x, and [N_2H_4] = 0.0085 - x: \(K_b = \frac{x^2}{0.0085-x}\) (c) Hydroxylamine: At equilibrium, [NH_3OH+] = x, [OH-] = x, and [NH_2OH] = 0.165 - x: \(K_b = \frac{x^2}{0.165-x}\)

Step 5: Determine the pH/pOH of the solutions

Calculate the [H+] or [OH-] concentration for each solution by solving the respective equilibrium constant expression. Then, find the pH or pOH as appropriate: (a) Hypochlorous acid: Calculate [H+], then find the pH: \(pH = -\log([H^+])\) (b) Hydrazine: Calculate [OH-], then find the pOH: \(pOH = -\log([OH^-])\) (c) Hydroxylamine: Calculate [OH-], then find the pOH: \(pOH = -\log([OH^-])\)

Step 6: Convert pOH to pH if needed

If you have calculated the pOH in Step 5, convert it to pH using the relationship: \(pH + pOH = 14\) (a) Hypochlorous acid: No conversion needed; the pH is already calculated. (b) Hydrazine: Convert the pOH to pH. (c) Hydroxylamine: Convert the pOH to pH. Following these steps, you will find the pH of each given solution.

Key Concepts

pH calculation

When solving for the pH of a solution, you're essentially measuring how acidic or basic the solution is. The pH scale ranges from 0 to 14. Pure water is neutral at a pH of 7. A value below 7 indicates an acidic solution, while a pH above 7 indicates a basic (alkaline) solution.

To determine pH:

• You have to know the concentration of hydrogen ions \([H^+]\) in the solution.
• You can easily calculate the pH by using the formula: \( pH = -\log([H^+]) \).

For basic solutions, you first find the pOH, which is related to the concentration of hydroxide ions \([OH^-]\).

• The formula for pOH is \( pOH = -\log([OH^-]) \).
• To find the pH from pOH, use the formula \( pH + pOH = 14 \).

Once you calculate these values, you'll know whether the solution is acidic or basic and by how much.

Equilibrium constant

The equilibrium constant, symbolized as \(K\), is crucial in understanding the extent to which a reaction can progress. It tells us about the concentrations of reactants and products in equilibrium for a reversible reaction. The general expression for an equilibrium constant is represented as:

• For acids, the symbol \(K_a\) is often used, indicating the acid's strength or how completely it dissociates in water to produce \[H^+\], such as in the ionization of hypochlorous acid \((HOCl)\).
• For bases, the symbol \(K_b\) is employed, indicating the base's strength or how completely it dissociates in water to yield \[OH^-\], like in the case of hydrazine \(N_2H_4\) and hydroxylamine \(NH_2OH)\).

The equilibrium expressions for these reactions are set up as:

• For acids: \( K_a = \frac{[H^+][A^-]}{[HA]} \)
• For bases: \( K_b = \frac{[BH^+][OH^-]}{[B]} \)

Where \[A^-\] and \[BH^+]\] are the conjugate bases and acids respectively. Knowing the equilibrium constants allows you to establish the formula needed to find concentrations of ions at equilibrium.

Dissociation reaction

Dissociation reactions are vital to calculating pH and understanding acid-base equilibrium. In these reactions, acids and bases break down in water to produce ions.

• An acid dissociates in water to release hydrogen ions (\([H^+]\)) and anion (e.g., hypochlorous acid dissociates to \(H^+\) and \(OCl^-\)).
• A base dissociates to yield hydroxide ions (\([OH^-]\)) and cations (such as hydrazine forming \(N_2H_5^+\) and hydroxylamine leading to \(NH_3OH^+\)).

Understanding these reactions helps in visualizing how the solution will behave. By setting up a dissociation equation, you essentially prepare a roadmap to forming equilibrium expressions. This allows you to begin solving for various concentrations and eventually, the pH.
When working with weak acids and bases, it helps to establish initial concentrations, as these compounds don't fully dissociate. Knowing the extent of dissociation helps when calculating the degree of reaction completion. This in turn aids in calculating the pH or pOH accurately.